The rebound of the body during uphill and downhill running ... · RESEARCH ARTICLE The rebound of the body during uphill and downhill running at different speeds A. H. Dewolf1,L.E.Peñailillo

RESEARCH ARTICLE

The rebound of the body during uphill and downhill running atdifferent speedsA. H. Dewolf1, L. E. Pen ailillo

2 and P. A. Willems1,*

ABSTRACTWhen running on the level, muscles perform as much positive asnegative external work. On a slope, the external positive and negativework performed are not equal. The present study analysed how theratio between positive and negative work modifies the bouncingmechanism of running. Our goals are to: (1) identify the changes inmotion of the centre of mass of the body associated with the slope ofthe terrain and the speed of progression, (2) study the effect of thesechanges on the storage and release of elastic energy during contactand (3) propose a model that predicts the change in the bouncingmechanism with slope and speed. Therefore, the ground reactionforces were measured on 10 subjects running on an instrumentedtreadmill at different slopes (from −9 to +9 deg) and different speeds(between 2.2 and 5.6 m s−1). Themovements of the centre of mass ofthe body and its external mechanical energy were then evaluated.Our results suggest that the increase in the muscular power iscontained (1) on a positive slope, by decreasing the step period andthe downward movements of the body, and by increasing the durationof the push, and (2) on a negative slope, by increasing the step periodand the duration of the brake, and by decreasing the upwardmovement of the body. Finally, the spring-mass model of running wasadapted to take into account the energy added or dissipated eachstep on a slope.

KEY WORDS: Locomotion, Running, Bouncing mechanism, Slope,External work

INTRODUCTIONWhen running on the level at a constant average speed, themechanical energy of the centre of mass of the body (COM)oscillates throughout the step like a spring-mass system bouncing onthe ground (Cavagna et al., 1976). During the rebound of the body,the muscle–tendon units (MTU) of the supporting lower limbundergo a stretch–shortening cycle during which part of themechanical energy of the COM is absorbed during the negativework phase to be restored during the next positive work phase(Cavagna et al., 1988).When running on the level, the upward and downward

movements of the COM are equal, and the positive and negativework done each step to sustain its movements relative to thesurroundings are equal (i.e.Wþ

ext ¼ W�ext). When running on a slope,

muscles are compelled to produce or dissipate energy to increase ordecrease the potential energy of the COM (DeVita et al., 2008). Inthis case, the spring-mass model is not suitable anymore because theratio between positive and negativework increases or decreases withthe slope of the terrain (Minetti et al., 1994). Several aspects of themechanics of human running uphill and downhill have been studiedthese last decades: e.g. the muscular work done, the energyconsumed and the muscular efficiency at different slopes (Minettiet al., 1994), the net muscular moment and power at the hip, kneeand ankle during uphill running (Roberts and Belliveau, 2005), andthe possible elastic energy storage and recovery in the arch andAchilles’ tendon while running uphill and downhill (Snyder et al.,2012).

To our knowledge, the change in the bouncing mechanism whilerunning on a slope at different speeds has never been analysed.Indeed, this mechanism will be affected by the ratio betweenpositive and negative work. Between 2.2 and 3.3 m s−1, this ratiochanges with the slope of the terrain, but not with the speed ofprogression (Minetti et al., 1994). However, to the best of ourknowledge, this ratio has never been measured while running on aslope at higher speeds.

To analyse the bouncing mechanism, we have measured the threecomponents of the ground reaction force (GRF) of 10 subjectsrunning on an inclined treadmill (from −9 to +9 deg, in 3 degincrements) at 10 different speeds (from 2.2 to 5.6 m s−1). Fromthese curves, we have analysed (1) the different periods of therunning steps, (2) the vertical movements of the COM and (3) theenergy fluctuations of the COM in order to understand how thebouncing mechanism of running deviates from the spring-massmodel with slope and speed. We believe that three aspects of thismechanism will be affected.

First, from a mechanical point of view, a spring-mass systembouncing vertically on the ground oscillates around an equilibriumpoint at which the vertical component of the GRF (Fv) is equal tobody weight (BW) (Cavagna et al., 1988). This is true whatever theslope; indeed, when running in steady state, the average verticalvelocity of the COM (V v) does not change from one step to the next.Consequently, the average vertical acceleration of the COM av ¼ 0and the average force Fv ¼ BW. The step period (T ) can thus bedivided into two parts: the first during which Fv>BW (tce), takingplace during the contact of the foot on the ground, and the secondduring which Fv≤BW (tae), taking place during both ground contactand the aerial phase. The period tce corresponds to the half period ofthe oscillation of the bouncing system. During tae, the bouncingmodel is not valid when the body leaves the ground.

When running on a flat terrain at speeds up to ∼3.1 m s−1, tce≈tae(symmetric step). As speed increases above 3.1 m s−1, tae becomesprogressively greater than tce (asymmetric step). This asymmetryarises from the fact that, when speed increases, the average verticalacceleration during tce (i.e. av;ce) becomes greater than theacceleration of gravity (g) whereas during tae, av;ae cannot exceedReceived 11 May 2016; Accepted 14 May 2016

1Laboratory of Biomechanics and Physiology of Locomotion, Institute ofNeuroScience, Universite catholique de Louvain, 1348 Louvain-la-Neuve, Belgium.2Exercise Science Laboratory, School of Kinesiology, Faculty of Medicine,Universidad Finis Terrae, Providencia, Santiago 7500000, Chile.

*Author for correspondence ([email protected])

P.A.W., 0000-0002-5283-4959

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1 g. Consequently, a longer tae is necessary to dissipate and restorethe momentum lost and gained during tce. At a given speed for agiven tce, the asymmetric step requires a greater av;ce; however, theincrease in the step period due to a longer tae results in a smallerinternal power necessary to reset the limbs at each step (Cavagnaet al., 1988).When running on a positive slope, muscles are performing more

positive than negative work. We expect that in order to contain theincrease in muscular forces during tce, the step remains symmetric

above 3.1 m s−1 and T becomes shorter than during running on thelevel. In contrast, when running on a negative slope, muscles areperforming more negative work. Because muscles are able todevelop higher forces during eccentric contractions, we expect thatT is tuned to contain the internal power, rather than the forceduring tce. Consequently, T should become longer than duringrunning on the level and the step should become asymmetric below3.1 m s−1.

Second, when running on a flat terrain, the momentums lost andgained over a step are equal, and V v=0. Consequently, the upwardsand downwards vertical displacements of the COM (S+ and S−) areequal. On the contrary, when running on a slope, Vv = 0 andS+≠S−. We expect that the dissimilarity between S+ and S− will bethe main factor affecting the bouncing mechanism. We also expectthat S− in uphill and S+ in downhill running will progressivelydisappear and with it the amount of energy that can potentially bestored in the MTU.

Third, on a flat terrain, the energy that is due to the verticalmovements of the COM (Ev) and the energy that is due to itshorizontal movements (Ef ) are fluctuating in phase (Cavagna et al.,1976), and the negative and positive work done are approximatelyequal: Δ+ Ef≈Δ− Ef and Δ

+ Ev≈Δ− Ev. However, when running on aslope, Δ+ Ef≈Δ− Ef but Δ

+ Ev≠Δ− Ev. We expect that (1) because ofthis imbalance, the fluctuations of Ef and Ev are no longer in phaseand, consequently, (2) an energy exchange occurs between thesetwo curves. Therefore, we have evaluated the duration of thepositive and negative work phases and the energy transductionbetween Ef and Ev during the period of contact tc (Cavagna et al.,2008a).

We hypothesize that the three aspects described here above willjeopardize the bouncing mechanism of running and in turn, thepossibility to store and release elastic energy into the elasticstructures of the lower limb. However, we also hypothesize that thedisappearance of this mechanism is intended to restrain the increasein Wþ

ext or W�ext that is due to slope.

Finally, we propose a model that describes the verticaloscillations of the COM during tce while running on a slope. Theaim of this model is to understand how all of the lower-limb musclesare tuned to generate the basic oscillation of the bouncing system.The spring-mass system bouncing on the ground that modelsrunning on the level (Blickhan, 1989; Cavagna et al., 1988) cannotdescribe running on a slope because energy must be added ordissipated. Therefore, we have incorporated an actuator in thespring-mass model that will generate a force proportional to Vv andproduce or absorb energy during contact.

MATERIALS AND METHODSSubject and experimental procedureTen recreational runners (three females and seven males) participatedin the study (age: 31.8±8.3 years, mass: 68.8±10.2 kg, height:1.78±0.07 m, mean±s.d.). Informed written consent was obtainedfrom each subject. The studies followed the guidelines of theDeclaration of Helsinki, and the procedures were approved by theEthics Committee of the Université catholique de Louvain.

Subjects ran on an instrumented treadmill at seven differentinclinations: 0, ±3, ±6 and ±9 deg. To neutralize the effect oflearning and muscle fatigue, half of the subjects started with aninclination of 0 deg that was increased and the other half with aninclination of 9 deg that was decreased. At each slope, subjects ranat 10 speeds presented in a different order (8, 10, 11, 12, 13, 14, 15,16, 18 and 20 km h−1, corresponding to 2.22, 2.78, 3.06, 3.33,3.61, 3.89, 4.17, 4.44, 5.00 and 5.56 m s−1, respectively). Note that

List of symbols and abbreviationsav vertical acceleration of the COMav average vertical acceleration of the COM over a complete

number of stepsav;ce, av;ae average vertical acceleration of the COMduring tce and taeb constant depending of the initial conditions in the

spring-actuator-mass modelBW body weightc actuator coefficient in the spring-actuator-mass modelCOM centre of mass of the whole bodyEext mechanical energy of the COMEf, El, Ev energy due to the fore-aft, lateral and vertical movement

of the COMEf,min, Ev,min minimum of the Ef– and Ev–time curvesFf, Fl, Fv fore-aft, lateral and vertical components of the GRFFp, Fn component of the GRF parallel and normal to the treadmill

surfaceFv average vertical GRF over a complete number of stepsGRF ground reaction forcek overall vertical stiffness of the leg-spring in the

spring-actuator-mass modelL step lengthMTU muscle–tendon unitRMSE root mean square errorS vertical displacement of the COMS+, S− upward and downward displacement of the COM over a

stepSþa , S

�a upward and downward displacement of the COM taking

place during taSþc , S

�c upward and downward displacement of the COM taking

place during tcSmin minimum vertical displacement necessary to overcome

the slope each stepT step periodtc, ta contact time and aerial timetce, tae effective contact time and effective aerial timetpush, tbrake duration of the positive and of the negative work phaseVbelt, Vbelt instantaneous and average velocity of the beltVf, Vf instantaneous and average fore-aft velocity of the COM

relative to the beltVl instantaneous lateral velocity of the COMVv, Vv instantaneous and average vertical velocity of the COM

relative to the beltWþ

ext, W�ext positive and negative external work done to sustain the

mechanical energy changes of the COM relative to thesurroundings

Wext sum of Wþext and the absolute value of W�

extWþ

f , W�f positive and negative work done to sustain the fore-aft

movement of the COMWþ

l , W�l positive and negative work done to sustain the lateral

movement of the COMWþ

v , W�v positive and negative work done to sustain the vertical

movement of the COM_W f , _W l , _W v power spent to move the COM in the fore-aft, lateral and

vertical directions%R percentage of energy recovered through the transduction

between Ev and Ef

θ slope of the treadmill

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for downhill running, the speed of the belt was limited by themanufacturer to 5.0 m s−1. At each slope, half the subjects startedwith the belt turning forwards to simulate uphill running and theother half with the belt turning backwards to simulate downhillrunning.Data were recorded during a period of 3–10 s, depending of the

speed and the inclination of the treadmill. Between 6 and 36 stepsper trial were recorded; a total of 15,760 steps were analysed. On a+6 deg slope, one subject could not run at 5.6 m s−1, and on a+9 deg slope, one subject could not run at 5.0 m s−1 and three couldnot run at 5.6 m s−1.

Experimental setup and data analysisThe instrumented treadmill (Fig. 1) consisted of a modifiedcommercial treadmill (h/p/Comos-Stellar, Germany, belt surface:1.6×0.65 m, mass: ∼240 kg) combined with four force transducers(Arsalis, Belgium), designed on the principle described by Heglund(1981). Because the whole body of the treadmill (including themotor) was mounted on the transducers, these were measuring the

three components of the GRF exerted by the treadmill under the foot(Willems and Gosseye, 2013): Fp, the component parallel to thelong axis of the tread surface; Fn, the component normal to the treadsurface; and Fl, the component in the lateral direction. The lowestfrequency mode of vibration was >41 Hz for Fp, 47 Hz for Fn and27 Hz for Fl. The non-linearity was <1% of full scale, and thecrosstalk <1%. The fore-aft (Ff ) and vertical (Fv) components ofGRF were then computed as:

ðFf Fv Þ ¼ cos u � sin usin u cos u

� �Fp

Fn

� �; ð1Þ

where θ is the angle between horizontal and the tread surface. Theelectrical motor was instrumented with an optical angle encoder tomeasure the speed of the belt (Vbelt). The average speed of the beltover a stride (V belt) differed by 2.8±1.4% (mean±s.d.) from thechosen speed and the instantaneous Vbelt did not change by morethan 5% of V belt.

0

1000

2000

-1

0

1

2

–1

0

1

Downhill 6 deg Level 0 deg Uphill 6 deg

3.3

3.5

3.7

Time (400 ms)

–0.5

0

0.5

a f (g

)

F f (N

)S

(m)

Vf (

m s

–1)

F v (N

)

BW a v (g

)

0.1

Vv

(m s

–1)

–400

0

400

Fig. 1. Schema of the instrumentedtreadmill (top) and typical time traces of asubject running at∼3.6 m s−1 on a−6 degslope (left column), on the level (middlecolumn) and on a +6 deg slope (rightcolumn). The schema at the top of the figurerepresents the instrumented treadmill. Thewhole body of the treadmill (including themotor) is mounted on four strain-gaugetransducers attached to wedges. Fp is thecomponent parallel to the long axis of thetreadmill surface, Fn the component normalto the tread surface and Fl the component inthe lateral direction (not represented here). θis the angle between horizontal and the treadsurface. The fore-aft (Ff ) and vertical (Fv)components of GRF are computed usingEqn 1. Traces from top to bottom. First andsecond row, left scale: vertical Fv andhorizontal Ff components of the groundreaction force exerted by the treadmill underthe foot; right scale: acceleration of the COM:av=(Fv−BW)/m and af=Ff/m, where BW is thebody weight and m the body mass. Thirdrow: vertical velocity of the COM, Vv, relativeto a referential attached to the treadmill belt.The horizontal dashed line represents theaverage vertical velocity of the COM over thestride (Vv ¼ Vbelt sin u). Fourth row: Vf is thefore-aft velocity of the COM. The dashedcurve represents the instantaneous velocityof the belt in the fore-aft direction (− Vbelt

cosθ). The horizontal dashed line representsthe average fore-aft velocity of the COM overthe stride (V f ¼ Vbelt cos u). Fifth row: verticaldisplacement of the centre of massS dividedinto contact phase (continuous line) andaerial phase (dotted line). Tracings wererecorded on a subject of height 1.83 m, bodymass 68.4 kg and age 24 years.

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The treadmill contained its own signal conditioning system:the GRF signals were amplified, low-pass filtered (4-pole Besselfilter with a −3 dB cut-off frequency at 200 Hz) and digitized bya 16-bit analog-to-digital converter at 1000 Hz. This system wasconnected to a PC via ethernet using TCP/IP (Genin et al.,2010). Acquisition and data processing were performed usingcustom-built software (LABVIEW 2010, National Instruments,Austin, TX, USA, and MATLAB 2013, MathWorks, Natick,MA, USA).

Division of the stepSteps were divided according to the Fv–time curves (Fig. 1): a stepstarted and ended when Fv became greater than BW. The effectivecontact time (tce) was the period during which Fv≥BW, and theeffective aerial time (tae) was the period during which Fv<BW(Cavagna et al., 1988). The time of contact (tc) was the periodduring which Fv>10 N, and the aerial phase (ta) was the periodduring which Fv≤10 N. The step duration was then calculated asT=tce+tae.

Measurement of the acceleration, velocity and verticaldisplacement of the COMThe acceleration (a), velocity (V ) and displacement (S) of the COMand the external work done (Wext) were computed from the GRFusing a method similar to that of Gosseye et al. (2010). Therefore,this method is only explained in brief.These computations were done over strides (i.e. two steps,

starting on the right foot). The fore-aft, lateral and verticalaccelerations of the COM relative to the reference frame of thelaboratory were calculated by dividing Ff, Fl and Fv � Fv by thebody massm (where Fv is the average vertical force over a stride). Intheory, Fv ¼ BW (as explained in the Introduction); therefore,strides were analysed only if Fv was within 5% of BW.The time curves of the three components of a were integrated

numerically to determine the fore-aft (Vf ) lateral (Vl) and vertical(Vv) velocity of the COM, plus an integration constant, which wasset on the assumption that the average velocity of the COM over a

stride was equal to V beltcosu for Vf, to zero for Vl and V belt sin ufor Vv.

The vertical displacement of the COM (S) was then computed bytime integration of Vv. The upward and downward displacements ofthe COM were divided according to tc (S

þc and S�c ) and ta (S

þa and

S�a ). The minimum vertical displacement (Smin) necessary toovercome the slope of the terrain each step can be computed bySmin ¼ VfT sin u, where V f is the average running speed over thestep and V fT is the step length.

Measurement of the positive and negative external workThe external work (Wext) is the work necessary to move the COMrelative to the surroundings plus the work done on or by theenvironment (Willems et al., 1995). It was measured from the workdone by the GRF (Gosseye et al., 2010).

The power spent to move the COM in the fore-aft ( _W f ), lateral( _W l) and vertical ( _W v) directions was computed, respectively, by:

_W f ¼ Ff Vf ; _W l ¼ Fl Vl and _W v ¼ Fv Vv: ð2Þ

In the fore-aft and vertical directions, the power spent by thesubject on the belt or by the belt on the subject can be computed,respectively, by:

� Ff vcosu and � Fvvsinu; ð3Þwhere v ¼ Vbelt � V belt represents the variation of Vbelt around V belt.Because these terms represented less than 3% of Wext, they wereneglected in this study.

The energies (Ef, El and Ev) that are due, respectively, to the fore-aft, lateral and vertical movements of the COM were computed by:

Ef ¼ð

_W f dt; E1 ¼ð

_W 1dt and Ev ¼ð

_W vdt; ð4Þ

and the total energy of the COM (Eext) by:

Eext ¼ðð _W f þ _W l þ _W vÞdt: ð5Þ

Uphill 3 deg

400 ms

Downhill 3 deg Downhill 6 deg Downhill 9 deg

Uphill 6 deg Uphill 9 deg

200

J

EfEv

Eext

Ef

Ev

Eext

Fig. 2. Energy changes of the COM during arunning stride. Mechanical energy–timecurves of the COM during a stride on differentslopes, while running at ∼4.2 m s−1. In eachpanel, the upper curve (Ef ) refers to the kineticenergy that is due to the forward motion of theCOM, the middle curve (Ev) to the sum ofthe gravitational potential energy (interruptedline) and the kinetic energy that are due to thevertical motion of the COM, and the bottomcurve (Eext=Ef+Ev) to the total energy of theCOM. The arrows pointing downwards indicatethe minimum of Ef and the arrows pointingupwards the minimum of Ev during the contactphase of the second step. The horizontalsegments of the energy curves correspond tothe aerial phase. Tracings are from the samesubject as in Fig. 1.

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The positive and negative work over one step (Wþf , Wþ

1 , Wþv ,

Wþext and W�

f , W�1 , W�

v , W�ext) were computed as the sum of the

positive and negative increments of the Ef, El, Ev and Eext

curves, respectively (Fig. 2). Because Wl represents less than1.5% of Wext, it is not presented separately in the results. Thework per step was computed as the work over the stride dividedby two.

Energy transduction between Ef and EvWhen running on the level, the Ef and Ev curves are in phase(Cavagna et al., 1976). However, when running on a slope, thesecurves could shift one relative to the other, allowing energytransduction between Ef and Ev (Fig. 2). The amount of energyrecovered (%R) over the contact phase was computed as (Cavagna

et al., 2008a):

%R¼100Wþ

f þjW�f jþWþ

v þjW�v jþWþ

1 þjW�1 j�ðWþ

extþjW�extjÞ

Wþf þjW�

f jþWþv þjW�

v jþWþ1 þjW�

1 j :

ð6Þ

Modelling the vertical movement of the body bouncingsystem of runningWhen running on the level, the vertical movement of the COMduring tce can be compared with the movement of a spring-masssystem bouncing vertically (Blickhan, 1989; Cavagna et al., 1988).

0

1

2 3 4 50

1

2

0

1

0

1

2

2 3 4 5 2 3 4 5 2 3 4 5 6

2 3 4 5

0

1

0

1

2 3 4 5 6

Uphill 3 deg Uphill 6 deg Uphill 9 deg

Downhill 3 deg Downhill 6 deg Downhill 9 deg

2 3 4 50

0.4

0.8

2 3 4 5 2 3 4 5 6

B

A

Uphill

Downhill

3 deg 6 deg 9 deg

Wex

t/Wex

t

W f+

Mas

s-sp

ecifi

c m

echa

nica

l wor

k pe

r uni

t dis

tanc

e (J

kg–

1 m

–1)

Average running speed (m s–1)

Wv+

Wv–

Wext+

Wext–

Wv+

Wv–

Wext+

Wext–

W f+

–

Fig. 3. Mass-specific external work per unitdistance as a function of speed at eachslope. (A) At each slope, the mass-specificpositive (closed symbols, superscript +) andnegative (open symbols, superscript −)mechanical work done each step is given as afunction of running speed. Wf is the work doneto accelerate or decelerate the COM (in thiscase, Wþ

f ¼ W�f ), Wv is the work to raise or

lower the COM and Wext is the muscular workactually done to sustain the movements of theCOM relative to the surroundings. Symbols andbars represent the grand mean of the subjects(n=10, except for +6 deg at 5.6 m s−1 and+9 deg at 5.0 m s−1 where n=9, and +9 deg at5.6 m s−1 where n=7) and the standarddeviations (when the length of the bar exceedsthe size of the symbol). In the middle row, thehorizontal continuous lines represent theminimumwork done to overcome the slope andthe dashed lines represent the work doneduring running on the level; these last lineswere drawn through the experimental data(weighted mean, Kaleidagraph 4.5). (B)W�

ext=Wext as a function of speed. The opendiamonds indicate negative slopes whereasthe closed diamonds indicate positive slopes.

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In this case, the balance of force is given by:

Fv � BW ¼ kS; ð7Þwhere k is the overall stiffness generated by the lower-limb muscles.To describe the vertical movement of the COM during running on aslope, Eqn 7 was implemented by incorporating an actuator parallelto the spring. This actuator generates a force proportional to themagnitude of Vv:

Fv � BW ¼ bþ kS þ cVv; ð8Þ

where b is a constant depending on the vertical velocity of the COMat touch down and c is the actuator coefficient. The coefficient c ispositive when running uphill and negative when running downhill.In this way, the power developed by the actuator force (cV 2

v ) ispositive in uphill running, showing that the actuator works like amotor giving energy to the body, and negative in downhill running,showing that the actuator works like a damper absorbing energy (seeinsets in Fig. 7A).At each instant i of tce (which represents the half-period of the

oscillation of the system), the vertical acceleration (av) of the massm

was computed by:

avðiÞ ¼ FvðiÞ � BW

m¼ b

m� k

mSðiÞ þ c

mVvðiÞ; ð9Þ

where av(i), Vv(i) and S(i) were the experimental data at instant i. Inthis way, n equations were produced, where n is the number ofsamples during contact. This set of equations resulted in an over-constrained system, from which the constant (b/m), the mass-specific stiffness (k/m) and the actuator coefficient (c/m) werecomputed by a regression analysis using singular valuedecomposition that minimizes the sum of squared errors.

For running on the level, we also compared the values obtainedfrom Eqns 7 and 8. In Eqn 8, c/m is close to zero and k/m differs by1.8±2.1% (mean±s.d., n=2440) from Eqn 7.

The goodness of the spring-actuator-mass model was assessed bycomputing (1) the variance explained by the regression model (r2),and (2) the root mean square error (RMSE), which expresses theagreement between the measured and the computed values of theGRF.

2 3 4 5 6

2 3 4 5 60

0.2

0.4

0

0.2

0.4

2 3 4 5 2 3 4 5

T

tce

tae

Uphill 3 deg Uphill 6 deg Uphill 9 deg

Tim

e (s

)

Downhill 3 deg Downhill 6 deg Downhill 9 deg

ta

T

0

1

2 3 4 5 2 3 4 5

0

1 av,ce

Uphill 3 deg Uphill 6 deg Uphill 9 deg

Downhill 3 deg Downhill 6 deg Downhill 9 degav,ce

A

B

Aver

age

verti

cal a

ccel

erat

ion

(g)

Average running speed (m s–1)

Fig. 4. Step duration and average verticalacceleration of the COM as a function of speedat each slope. (A) At each slope, the filled squaresindicate the step period (T ), the open triangles theeffective contact time (tce) and the closed trianglesthe effective aerial time (tae). The red lines representthe aerial time (ta). The dashed lines are the resultsobtained on the level for T, tce, tae and ta (weightedmean, Kaleidagraph 4.5). (B) At each slope, theopen triangles indicate the average verticalacceleration of the COM during the effective contacttime (av;ce). Dashed lines are the results obtainedon the level and the horizontal dotted linerepresents 1 g. Other indications as in Fig. 3.

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StatisticsData were grouped into speed–slope classes. To obtain one valueper subject in each class, the steps of the same subject in a sameclass were averaged. The mean and standard deviation of the runnerpopulation were then computed in each class (grand mean). A two-way ANOVAwith post hoc Bonferroni correction (PASW Statistics19, SPSS, IBM, Armonk, NY, USA) was performed to assess theindividual and interaction effects of speed and slope on thecalculated variables (P-values were set at 0.05).

RESULTSEffect of slope on the ratio between positive and negativework doneThe mass-specific external work done per unit distance is plotted inFig. 3A. In the fore-aft direction, the work done to move the COMhorizontally (Wf ) increases similarly with speed at any slope(P=0.995). Moreover, because subjects move at a constant speed,Wþ

f ¼ W�f .

Therefore, the imbalance betweenWþext andW

�ext on a slope is only

due to a modification in Wþv and W�

v . During uphill running, Wþv

increases, while W�v decreases and tends to disappear above

∼4.4 m s−1 at +6 deg and above∼3.3 m s−1 at +9 deg. On a negativeslope,Wþ

v tends to disappear above∼5.0 m s−1 at−6 deg and above∼4.2 m s−1 at −9 deg.

Note that the ratioW�ext=Wext changes with slope (P<0.001), but is

independent of speed (P=0.17): W�ext=Wext equals 0.5 at 0 deg, but

decreases monotonically below 0.5 on a negative slope andincreases monotonically above 0.5 on a positive slope (Fig. 3B).

Effect of slope and speed on step periodThe step period (T ) is given as a function of running speed inFig. 4A. The slope has a significant effect on the step period(P<0.001): T decreases when running uphill, whereas it increaseswhen running downhill. The effect of slope on T is more markedon positive than on negative slopes, though this effect issignificant at all slopes (Bonferroni post hoc, P<0.01). The slopedoes not affect the effective contact time tce, except at +9 deg(Bonferroni post hoc, P<0.01). Thus, the change in T is mainlydue to a change in tae (P<0.001), which in turn, is largely due toa change in ta.

When running on the level, the step is symmetric (i.e. tae≈tce) atspeeds up to ∼2.8 m s−1 (Bonferroni post hoc, P<0.01). Theseresults are consistent with those reported in the literature (Cavagnaet al., 1988; Schepens et al., 1998). When running uphill, tae≈tce ina larger range of speeds because the mean vertical accelerationduring tce (av;ce) is kept smaller than 1 g at higher speeds than on thelevel (Fig. 4B): up to ∼3.3 m s−1 at +3 deg, to ∼3.9 m s−1 at +6 degand to 5.6 m s−1 at +9 deg. When running downhill, the difference

2 3 4 5 2 3 4 5 62 3 4 50

0.1

0.2

0

0.1

0.2

0

0.1

0

0.1

2 3 4 5 2 3 4 5 2 3 4 5 6

S+

Uphill 3 deg Uphill 6 deg Uphill 9 deg

Downhill 3 deg Downhill 6 deg Downhill 9 deg

S–

S–

S+

Sa–

Sc

Sa +

Sc +

Verti

cal d

ispl

acem

ent o

f the

CO

M (m

)

Average running speed (m s–1)

–

Fig. 5. Vertical displacement of the COM as afunction of speed at each slope. At each slope,the filled squares indicate the upward (S+, upperpanel) and downward (S−, bottom panel) verticaldisplacement of the COM. The grey zone (Sa)represents the fraction of the vertical displacementtaking place during the aerial phase, whereasSc isthe vertical displacement taking place during thecontact phase. The dashed lines indicate S duringlevel running. The continuous line representsSmin,the minimum vertical displacement to overcomethe slope during the step. Note thatSmin increaseswith speed because the step length increases.Other indications as in Fig. 3.

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between tae and tce tends increase because av;ce becomes >1 g atspeeds lower than those on the level. Consequently, the range ofspeeds at which tae≈tce becomes narrower, i.e. below ∼2.2 m s−1 at−6 and −9 deg.

Effect of slope and speed on the upward and downwarddisplacements of the COMOn the level, the upward (S+) and downward (S−) displacements ofthe COM over one step are equal (Fig. 5). On a slope, the differencebetween S+ and S− increases with inclination (P<0.001).When running uphill, S+ increases with slope and speed. At slow

speeds, the displacement upwards during the aerial phase (Sþa ) isalmost nil, whatever the slope. When speed increases, Sþa representsa greater part of S+. The displacement downwards (S−) decreaseswhen slope and speed increase: S− disappears above 5.0 m s−1

at +6 deg and above 3.9 m s−1 at +9 deg. Note that the reduction ofS− is first due to a reduction of S�a . At +6 deg and +9 deg, S�a isalmost nil at all speeds, suggesting that Vv at touchdown is close tozero.

When running downhill, the opposite phenomenon is observed:S− increases and S+ decreases with slope and speed. The effect ofslope and speed is more marked on S− when running downhill thanon S+ when running uphill. When running downhill, S�a represents asignificant part of S−, whereas Sþa is almost nil whatever the slopeand speed, indicating that Vv at take-off is close to zero.

Effect of slope and speed on the energy fluctuations of theCOMWhen running on a slope, muscles are compelled to modify the ratiobetween concentric and eccentric contraction. Consequently, thetiming of negative and positive work production during tc ismodified (Fig. 6A). The time during which muscles performnegative external work (tbrake) is extended during downhill running(P<0.001) while the duration of the positive work phase (tpush) isincreased during uphill running (P<0.001). Because at a givenspeed, tc does not change significantly with slope (Fig. 6A),the modification of tbrake is compensated by an opposite changein tpush.

–9 –6 –3 0 3 6 9–9 –6 –3 0 3 6 90

0.1

0.2

0.3

–9 –6 –3 0 3 6 9

-9-6-30369

Begin tc Ev,min Ef,min End tc

0 10 20–9–6–30369

–50 0 50 100 150

–9–6–30369

5.0 m s–1

2.2 m s–1

3.6 m s–1

Recovery (%)Time (% of tc,e)

Slo

pe (d

eg)

Tim

e (s

)

B C

A

tctpush

tbrake

Slope (deg)

2.2 m s–1 5.0 m s–13.6 m s–1 Fig. 6. Contact time and energy transductionbetween Ef and Ev during the contact phase as afunction of slope at low, intermediate and fastspeeds. (A) In each panel, the contact time (tc, filledsquares) is plotted as a function of slope and isdivided into the time during which muscles performpositive external work (tpush, filled circles) and the timeduring which muscles perform negative external work(tbrake, open circles). The vertical dashed linescorrespond to running on the level. The dotted linesare drawn through the data (weighted mean,Kaleidagraph 4.5). (B) The abscissa represents therelative time expressed as a percentage of tce and theordinate represents the different slopes studied. Filledand open squares correspond, respectively, to thetouchdown and take-off. The vertical dashed linesrepresent the beginning and end of tce. Note that theperiod between touchdown and the beginning of tceand the period between the end of tce and take-offchange little with slope. Both the Ef and the Ev curvesdecrease during the first part of contact, and increaseduring the second part (Fig. 2). The open and filledsymbols represent, respectively, the time at which theEv and Ef curves reach a minimum (Ev,min and Ef,min).The grey zone illustrates the period during which anenergy transduction between Ev and Ef is possiblebecause one curve is still decreasing while the otherhas begun to increase. Note that the instant of Ef,min

changes little with slope. As compared with levelrunning, Ev,min appears earlier in the stride on positiveslopes and later on negative slopes. (C) The amountof energy recovered during the phase encompassedbetween the two minima of Ef and Ev, computed byEqn 6, is presented at each slope for the threespeeds. The horizontal dashed lines correspond torunning on the level. Other indications as in Fig. 3.

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The relative time of occurrence of the minimum of Ef (Ef,min)changes little with slope [see Figs 2 (arrows) and 6B]. On thecontrary, the minimum of Ev (Ev,min) appears relatively earlierduring contact in uphill running and later in downhill running. Theeffect of the slope on the relative time of occurrence of Ev,min ismore accentuated when speed increases. Between Ef,min andEv,min, the Ef and Ev curves (grey zone in Fig. 6B) are out of phaseand an energy exchange can occur from Ef to Ev when Ev,min

precedes Ef,min, and from Ev to Ef when Ef,min precedes Ev,min.This exchange of energy allows recovery of a significant amountof energy (i.e. %R>10%) only on steep slopes and at speeds>3.6 m s−1 (Fig. 6C).

Modelling running on a slopeWhen running on a slope, the lower-limb muscles can be modelledas a mass mounted on a spring in parallel with an actuator thatgenerates a muscular driving force proportional to the verticalvelocity of the COM and produces (uphill) or absorbs (downhill)energy during tce. The overall mass-specific stiffness k/m and theactuator coefficient c/m generated by the lower-limb muscles(Fig. 7B) are modified by both the slope of the terrain and the speedof progression (P<0.001). Compared with at 0 deg, k/m decreases ona positive slope and increases on a negative slope. At a given slope,k/m increases with velocity, though the effect of speed is greater onnegative than positive slopes (P<0.001).

During level running, the complex musculoskeletal system of thelower limb behaves like a single linear spring and c/m is nil.Running uphill requires additional energy to overcome the slope. Inthis case, c/m is positive and increases with the mechanicaldemands, i.e. with increasing slope but also with increasing speed.On the contrary, running downhill requires dissipation of energy.Consequently, c/m is negative and its absolute value increases whenslope becomes steeper and when speed increases. Note also that theeffect of speed on c/m is greater on negative than on positive slopes(P<0.001).

This model describes the basic vertical oscillation of the COMduring running on a slope (Fig. 7A) without taking into account thefirst peak in the Fv curve as a result of foot slap (Alexander et al.,1986; Schepens et al., 2000) and the vibrations of the treadmillmotor (Fig. 1); the r2 of the least square method is always >0.63 andRMSE ranges between 120 and 440 N (Table 1).

DISCUSSIONThis study was intended to help understand how the bouncingmechanism is modified when the slope of the terrain becomespositive or negative. Our results show that the bouncingmechanism still exists on shallow slopes and progressivelydisappears when the slope increases. This mechanism disappearsearlier on positive than on negative slopes and earlier at highspeeds than at slow speeds. In this section, we will discuss how

100

400

700

2 3 4 5–40

–20

0

20

2 3 4 5 2 3 4 5 6

Uphill

Downhill

3 deg 6 deg 9 deg

0 0.04 0.08

R2=0.87RMSE=198

0 0.04 0.08

R2=0.88RMSE=186

0

1000

2000

0 0.04 0.08

R2=0.83RMSE=238

Downhill 6 deg Level 0 deg Uphill 6 degVe

rtica

l for

ce (N

)

BW

Uphill

Downhill

A

B

Vertical displacement of the COM (m)

Average running speed (m s–1)

Verti

cal s

tiffn

ess

(s–2

)D

ampi

ng c

oeffi

cien

t (s–

1 )Fig. 7. Mechanical model of running on aslope. (A) Typical trace of the vertical force Fv

during the contact period (tc) plotted as a functionof vertical displacement S during running at∼3.6 m s−1. The left panel corresponds torunning at −6 deg, the middle panel to 0 deg andthe right panel to +6 deg. The upwards arrowindicates the downward movement of the COMtaking place during the first part of tc; thedownwards arrow indicates the upwardmovement of the COM taking place during thesecond part of tc. The black dashed linecorresponds to Fv=BW. The red dashed linecorresponds to the predicted value of Fv

computed using Eqn 9 during tce using the valuesof b, k and c obtained by the regression analysis.Tracings are from amale subject (height: 1.83 m,body mass: 70.0 kg, age: 31 years). The r2 andRMSE values are indicated on each trace. Theinsets illustrate the model used in downhill (aspring in parallel with a damper), level (a singlespring) and uphill (a spring in parallel with amotor) running. (B) The upper row presents themass-specific vertical stiffness k/m as a functionof speed, and the lower row the mass-specificcoefficient c/m of the actuator. The left columncorresponds to a slope of 3 deg (open circles fordownhill running and filled circles for uphillrunning), the middle column to a slope of 6 degand the left column to 9 deg. Dashed linescorrespond to level running. The dotted lines aredrawn through the data (weighted mean,Kaleidagraph 4.5). Other indications as in Fig. 3.

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the step period, the vertical movement of the COM and theenergy fluctuation of the COM affect the bouncing mechanism ofrunning. We will also show that these variables are tuned tocontain the increase in the positive or negative muscular workand power that is due to slope. We finally discuss the quality andlimits of the mechanical model describing the vertical movementof the COM during slope running.

Change in step period with slopeMinetti et al. (1994) show that on a positive slope the step period Tdecreases significantly, whereas on a negative slope T tends toincrease (though this increase is not significant). Our results confirmthat, at a given speed, T is shorter when running uphill than whenrunning on the level. On the contrary, during downhill running, ourresults show a significant increase in T, even if this change is lessmarked than on a positive slope. This discrepancy with the results ofMinetti and colleagues may be explained by the fact that the numberof steps analysed here (n=15,760 steps) is higher than that inMinetti’s study (n=418 steps).In our study, we observe that tc (Fig. 6A) and tce (Fig. 4A) at a

given speed do not change significantly with slope, in both downhilland uphill running (except at +9 deg at the highest speeds). Thechange in the step period T is thus mainly due to a change in ta,which in turn changes tae.As shown by Cavagna et al. (1991), running on flat terrain with a

long effective aerial phase is a convenient strategy to decrease theaverage power developed over the step, provided that muscles areable to develop enough power during the push. Our results suggestthat the changes in tae are intended to contain the additionalmuscular power due to the slope.When running uphill as compared with running on the level, T

decreases because tae is reduced. Because tce is not modified byslope, the step remains symmetric (i.e. tce≈tae) at higher speeds thanon flat terrain. These results are similar to those obtained whenrunning on the level in hyper-gravity (Cavagna et al., 2005): at 1.3 g,the rebound remains symmetric up to 4.4 m s−1.At a given speed and for a given tce, a symmetric rebound results

in a shorter step than does an asymmetric rebound. As a result, theminimal height (Smin) that the COM must gain each step is smaller,the impact against the ground is reduced and the force and power

during the push are decreased (Cavagna et al., 1991). However,decreasing T results in a greater internal power to move the limbsegments relative to the COM.

Snyder and Farley (2011) have observed that the optimal stepperiod at which the oxygen consumption is minimal does notchange for slopes between −3 and +3 deg. However, these authorsshow that the freely chosen T decreases slightly between 0 and3 deg. This decrease in T might be a strategy to contain themechanical power during the push.

The changes observed in uphill running are similar to thoseobserved in old men running (Cavagna et al., 2008b). In oldersubjects, the average upward acceleration (av;ce) is lower than inyounger ones, leading to a symmetric rebound. According to theseauthors, the lower force attained during contact by the old subjectsmay be explained in part by the loss of muscular strength (e.g.Doherty, 2003). Similarly, the strategy adopted while running uphillcould be due to the limits set by the muscular strength and/or power.Furthermore, in uphill running, the GRF vector is farther from theleg joint centres (DeVita et al., 2007). Thus, the slope of the terrainalters the muscle mechanical advantage by creating longer leverarms, and leads to higher joint torques and power outputs (Robertsand Belliveau, 2005). Therefore, decreasing Tmight be a beneficialstrategy to limit the muscular moments and to reduce the mechanicalload applied to the MTU.

At the opposite, when running downhill, T increases as comparedwith running on the level because tae increases. Because tce is notmodified, steps are asymmetric (i.e. tce<tae) at lower speeds than on aflat terrain. These results are similar to those obtained when runningin unweighted conditions (Sainton et al., 2015): when BW isreduced by 40%, tc remains unchanged while ta is increased.

The step asymmetry has the physiological advantage to limit theinternal power. However, this asymmetry requires a greater av;ceand, consequently, a greater power during tce. The choice of thisstrategy in downhill running may be due to the difference in forceexerted during negative and positive work phases: during downhillrunning, muscles contract mainly eccentrically, allowing thedevelopment of higher forces. Moreover, in downhill running,runners land with the leg more extended (Leroux et al., 2002),resulting in reduction of the lever arms of the GRF about the lower-limb joints and thus of the net muscular moments.

Table 1. Coefficient of determination and root mean square error (RMSE) of the model

Speed (m s−1)

Slope (deg) 2.2 2.8 3.1 3.3 3.6 3.9 4.2 4.4 5.0 5.6

9 0.80±0.04217±45

0.80±0.04232±52

0.78±0.04246±51

0.77±0.05250±55

0.76±0.05254±54

0.75±0.05259±49

0.74±0.07269±53

0.72±0.05280±46

0.70±0.04280±52

0.66±0.06307±37

6 0.86±0.03182±35

0.87±0.03188±42

0.87±0.03194±43

0.86±0.03206±43

0.85±0.02210±43

0.84±0.03226±48

0.83±0.04234±48

0.82±0.04242±51

0.77±0.03276±50

0.75±0.05295±54

3 0.93±0.01139±29

0.92±0.01146±31

0.92±0.02155±33

0.91±0.02167±35

0.90±0.02178±28

0.89±0.02189±33

0.87±0.02209±29

0.86±0.03216±33

0.82±0.06258±43

0.75±0.1312±52

0 0.94±0.02120±29

0.92±0.03153±37

0.90±0.05175±48

0.87±0.04200±37

0.86±0.06210±39

0.84±0.05238±49

0.80±0.07264±51

0.78±0.07291±55

0.72±0.1347±71

0.68±0.1400±89

−3 0.93±0.05125±40

0.93±0.02142±28

0.92±0.02153±33

0.91±0.02170±33

0.90±0.03188±44

0.88±0.04203±34

0.86±0.04230±49

0.84±0.04251±48

0.77±0.07308±66

−6 0.81±0.2207±98

0.86±0.07207±60

0.86±0.04211±43

0.84±0.04234±51

0.83±0.04248±55

0.82±0.05269±66

0.8±0.04289±60

0.77±0.04311±60

0.72±0.06369±64

−9 0.77±0.1248±63

0.75±0.1292±62

0.76±0.09297±64

0.73±0.08330±84

0.71±0.11339±88

0.70±0.08357±75

0.70±0.06372±73

0.68±0.06392±82

0.63±0.1444±91

The mass-specific stiffness (k/m) and damping/motor coefficient (c/m) during tce were evaluated using a least-squares method. In each class, the first linecorresponds to the r2 and the second to the RMSE (expressed in N).Data are grand means (see Materials and methods)±s.d. In each class, n=10 except for (+6 deg; 5.6 m s−1) and (+9°; 5.0 m s−1), where n= 9, and for (+9 deg;5.6 m s−1), where n=7.

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Change in vertical motion of the COMWhen running uphill at a given speed, to maintain the bouncingmechanism similar to level running, the runners should increaseWþ

extto overcome the slope, without changing W�

ext (Fig. 3A). However,the runner limitsWþ

ext by reducingW�ext. The opposite phenomenon is

observed during downhill running, whereW�ext is limited by reducing

Wþext. Because the work done to accelerate and decelerate the COM

forwards (Wf ) does not change with slope, the change in Wþext and

W�ext is due to a change inW

þv andW�

v , which in turn is mainly due toa modification of S+ and S− during the step (Fig. 5).When running uphill, the upward displacement of the COM each

step (S+) increases with slope and speed because Smin increases. Incontrast, the downward displacement (S−) decreases. At low speedsand on shallow slopes, S− is still present because the muscularpower at disposal during the push is great enough to increase S+

beyond Smin. In this way, the presence of S− allows the storage ofelastic energy into the MTU to be restored during the next positivework phase. On the contrary, at high speeds and on steep slopes, thepower during the push approaches the maximal performance of therunner. Therefore, S+ is maintained close to Smin; consequently, S

−

and W�v (Fig. 3A) are almost nil and no rebound is possible.

When running downhill, S− increases faster with slope and speedthan S+ in uphill running (Fig. 5); these changes in S− are mainlydue to a change in the downward fall of the COM during ta (S

�a ). In

contrast, the effect of slope and speed on S+ on a negative slope isless marked than the changes in S− on a positive slope.Consequently, when running downhill, the possibility of elasticstorage increases with speed and slope. Though, as mentioned bySnyder and Farley (2011) and Snyder et al. (2012), the recoil islimited because S+ and Wþ

v are small.Furthermore, the large S�a observed in downhill running results in

a high Vv at touchdown, which in turn causes greater GRF.According to Zelik and Kuo (2012), the vibrations of the soft tissuesinduced by the impact could reduce the muscular work done bydissipating energy.

Change in the energy fluctuations of the COMRunning is thought to employ a spring-mass mechanism by whichinteractions between the COM and the ground allow storage andrelease of elastic energy in the MTU. In vivo measurements ofmuscle–tendon interaction have highlighted the influence of thestretch–recoil of tendons on the power output of muscles duringrunning (Ishikawa and Komi, 2008; Roberts et al., 2007).On the level at low and intermediate speeds, tpush>tbrake (Fig. 6A),

although Wþext ¼ W�

ext. This difference in time is due to the greatermuscular force exerted during the eccentric phase. The fact thattpush>tbrake thus suggests an important contribution of the contractilemachinery to the MTU length change and to the work production(Cavagna, 2006). At higher speeds (i.e. above ∼3.9 m s−1),tpush=tbrake, which suggests that when muscle activation isprogressively augmented with increasing speed, the MTU lengthchange is mainly due to tendon length change. Thework contributionby the contractilemachinery is thusprogressivelysubstitutedbyelasticstorage and recovery by tendons. The tpush/tbrake ratiomay therefore bean expression of the deviation of the MTU’s response from that of anelastic structure. On a slope, the change in the ratio betweenWþ

ext andW�

ext most likely affects the interaction between muscle and tendonsduring the stretch-shortening cycle (Roberts and Azizi, 2011).When running uphill, the linear increase of Wþ

ext changes thepartitioning of tc into tpush and tbrake (Fig. 6A). In order to contain theaverage muscular power required during the push, tpush increaseswith slope and tpush>tbrake. This suggests that the MTU length

change is mainly due to a shortening of the contractile machineryand that less elastic energy is stored – as in old men running on thelevel (Cavagna et al., 2008a). As proposed by Roberts and Azizi(2011), it could be that on a positive slope, MTUs work like poweramplifiers: the energy produced during the muscular contraction isstored at a low pace in the tendons to be released at a higher pace.

In downhill running when slope becomes steeper, in order to limitthe power during the brake, tbrake increases and tpush becomesshorter. The higher GRF observed in downhill running could favourthe role of the tendon relative to that of muscle: muscle wouldperform a quasi-isometric contraction and the energy would bestored in the tendon during rapid stretch, to be dissipated later by aslower lengthening of the muscle (Roberts and Azizi, 2011).

The partition between tpush and tbrake changes because theminimum of Ev appears increasingly earlier in uphill running andincreasingly later in downhill running, whereas the minimum of Ef

occurs always more or less in the middle of the contact period(Fig. 6B and arrows in Fig. 2). For this reason, when slope increases,a phase shift between Ef and Ev emerges and an energy transductionbetween these two forms of energy can occur.

When running uphill, the lower limb acts like a pole in athletics(Schade et al., 2006). During the first part of contact (i.e. periodbetween the two arrows in the upper panels of Fig. 2), the COMloses horizontal velocity, while it gains height and vertical velocity.During this phase, there is an energy transduction between Ef andEv. The fact that Eext decreases shows that the loss in Ef is greaterthan the gain in Ev; part of the Eext lost is stored in the elastic elementof the MTU to be released during the second part of contact toincrease the kinetic and potential of the COM. Note that the samephenomenon during the running step preceding the jump over anobstacle was described by Mauroy et al. (2013).

When running downhill, the energy transduction occurs duringthe second part of the contact, when Ev is decreasing while Ef

increases. In this case, the potential energy lost is used to acceleratethe COM forward.

In level running, the energy recovered through an exchangebetween Ev and Ef (Eqn 6) is negligible (%R<5%), like in a spring-mass system (Cavagna et al., 1976). When running at high speed onsteep slopes, the rebound of the body deviates from a spring-masssystem. However, the energy transduction allows recovering only asmall amount of energy: at best %R≈20% in uphill running and%R≈10% in downhill running.

Model of the vertical bounce of the bodyIn this study, we propose a model that reproduces in firstapproximation the basic oscillation of a spring-mass actuator (seeinset in Fig. 7A). The aim of this model is to understand how all ofthe lower-limb muscles are tuned to generate an overall leg stiffnessand leg actuation to sustain the vertical oscillation of the bouncingsystem.

Running on a slope is not a pure elastic phenomenon becauseenergy must be added or released at each step. Therefore, our modelincludes an actuator placed parallel to the spring. This actuatorgenerates a force proportional to Vv; a linear force–velocityrelationship was chosen because functional tasks that involve alllower-limb joints show a quasi-linear relationship (Bobbert, 2012;Rahmani et al., 2001), rather than the classical hyperbolicrelationship described in isolated muscles (Hill, 1938). Thecoefficient c/m is positive during uphill running, showing that theactuator adds energy to the system, and negative during downhillrunning, because energy is dissipated at each step. As speed andslope increase, the discrepancy between the upward and downward

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displacement of the COM increases, the rebound of the COMdeviates from a spring-mass system and the coefficient c/m increases.On a slope, k/m decreases when running uphill and increases

when running downhill. On a positive slope, in order to maintain thesymmetry of the step, av;ce remains ≤1 g (Fig. 4B). This loweracceleration is most likely obtained by reducing the stiffness of thecontractile elements of the MTU and the overall k/m becomessmaller than that on the level (Fig. 7B).Note also that on positive slopes, the limbs at impact are more

flexed to prevent pitching backward (Birn-Jeffery and Higham,2014; Leroux et al., 2002). On the level, runners adopting a morecrouched posture, similar to Groucho running (McMahon et al.,1987), present lower vertical GRF and consequently a smaller k/m.In uphill running, the more flexed posture during stance couldexplain, at least in part, the smaller k/m.On a negative slope, the step is asymmetric at most speeds and

slopes because av;ce>1 g (Fig. 4B). When acceleration increases,muscles fibres oppose a progressively greater force to stretching andthe contractile machinery becomes stiffer than the tendon.Subsequently, the overall k/m becomes greater than on the level.Furthermore, increasing k/m minimizes the lowering of the COMduring contact (Fig. 5). This strategy might serve as an intrinsicsafety mechanism to limit the risk of MTU damage after landing(DeVita et al., 2008).Our simple spring-actuator-mass model predicts the Fv–time

curve, though it does not take into account the first peak in the Fv

curve, which is due to foot slap (Alexander et al., 1986; Schepenset al., 2000). To estimate the contribution of the foot collision on theshape of the Fv–time curve, we used a Fourier series analysis (Clarkand Weyand, 2014). This analysis decomposes the Fv signal intolow- and high-frequency components. According to Clark andWeyand (2014), the high-frequency components are mainly due tothe acceleration of the lower limb during the impact phase. In ourstudy, we observed that the variance accounted for by these high-frequency components increases at high speeds and on steepnegative slopes (see Table S1). This observation supports the ideathat the decrease in goodness-of-fit indexes (Table 1) is due to ahigher contribution of the foot’s interaction with the ground.These last results corroborate those of Clark and Weyand (2014)

obtained in sprint running; they show that at swift speeds, the GRFwaveform deviates from the simple spring-mass model pattern, mostlikely because of the greater importance of the foot–groundcollision. Therefore, Clark and Weyand (2014) propose a modelincluding two masses and springs to take into account theinteraction of the lower-limb segments with the ground. Ourspring-actuator-mass model could thus be refined by adding asecond spring mass representing the foot and shank, althoughkinematic data of these segments are needed to feed thisimplemented model.

ConclusionsCavagna et al. (1991) suggested that running with a long aerialphase limits the step-average power as long as muscles are able todevelop enough power during the push and/or the brake. Our resultssupport this hypothesis.When running uphill at a given speed, the average external power

developed during the positive work phase seems to be the limitingfactor. Actually, when slope increases, in order to keep a long aerialtime ta, the vertical velocity of the COM at take-off should increasebecause the minimal vertical displacement (Smin) increases. Thiswould require a greater power during the push. As a matter of fact,this power is limited by: (1) reducing ta and thus the step period T,

(2) increasing the duration of the push (tpush) at the expense of theduration of the brake (tbrake) and (3) reducing the downwarddisplacement of the COM. As a result, uphill running deviates froma bouncing mechanism as speed and slope increases.

When running downhill, the average external power developedduring the negative work phase seems to be the limiting factor.Indeed, despite a lower vertical velocity at take-off, ta – and thus T –increases with slope and speed because the ballistic fall of the COMincreases. A longer ta increases the external power developed duringthe brake because the energy to be dissipated after touchdown isgreater. In spite of better muscular performance during eccentricthan concentric contraction, the power during the brake is limited by(1) increasing tbrake at the expense of tpush and (2) reducing theupward displacement of the COM. Consequently, the bouncingmechanism during downhill running gradually disappears as speedand slope increase.

AcknowledgementsThe authors thank Profs D. De Jaeger and V. Legat for their advice.

Competing interestsThe authors declare no competing or financial interests.

Author contributionsConceived and designed the experiments: A.D., P.L., P.W. Performed theexperiments: A.D., P.L., P.W. Analyzed the data: A.D., P.L., P.W. Wrote the paper:A.D., P.L., P.W.

FundingThis study was funded by the Universite catholique de Louvain (Belgium), theUniversidad Finis Terrae (Chile) and the Fonds de la Recherche Scientifique(Belgium).

Supplementary informationSupplementary information available online athttp://jeb.biologists.org/lookup/doi/10.1242/jeb.142976.supplemental

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